Hypersurface-orthogonal generators of an orthogonally transitive transitive $G_2I$, topological identifications, and axially and cylindrically symmetric spacetimes

TL;DR

This paper revisits a criterion for hypersurface-orthogonal generators in spacetimes with a $G_2I$ symmetry, exploring local and global metric forms, topological effects, and physical implications, especially for cylindrical and toroidal symmetries.

Contribution

It reformulates a known criterion as a test for metric properties, analyzes topological and global aspects of $G_2I$ spacetimes, and clarifies the physical significance of topological parameters.

Findings

Hypersurface-orthogonal $G_2I$ generators imply diagonalizable or null-Killing vector metrics.

Topological identifications introduce additional parameters affecting global geometry.

These parameters influence physical source characteristics and holonomy in cylindrical and toroidal spacetimes.

Abstract

A criterion given by Castejon-Amenedo and MacCallum (1990) for the existence of (locally) hypersurface-orthogonal generators of an orthogonally-transitive two-parameter Abelian group of motions (a $G_2I$) in spacetime is re-expressed as a test for linear dependence with constant coefficients between the three components of the metric in the orbits in canonical coordinates. In general, it is shown that such a relation implies that the metric is locally diagonalizable in canonical coordinates, or has a null Killing vector, or can locally be written in a generalized form of the `windmill' solutions characterized by McIntosh. If the orbits of the $G_2I$ have cylindrical or toroidal topology and a periodic coordinate is used, these metric forms cannot in general be realized globally as they would conflict with the topological identification. The geometry then has additional essential parameters, which specify the topological identification. The physical significance of these parameters is shown by their appearance in global holonomy and by examples of exterior solutions where they have been related to characteristics of physical sources. These results lead to some remarks about the definition of cylindrical symmetry.